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Springer eBooks

Critical Points in Two-Dimensional Replica Sigma Models

Paul Fendley

Published Jan 1, 2001• 6 citations
Publisher
DOI
10.1007/978-94-010-0838-9_6

Related papers

Critical points in two-dimensional replica sigma models

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I survey the kinds of critical behavior believed to be exhibited in two-dimensional disordered systems. I review the different replica sigma models used to describe the low-energy physics, and discuss how critical points appear because of WZW and theta terms.

arXiv:cond-mat/0006360•Jun 23, 2000•1 citation

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We study a completely-packed loop model with crossings in a three-dimensional lattice and confirm it is described by $\mathrm{RP}^{n-1}$ sigma field theories. We use Monte Carlo simulations, with systems sizes up to $1800\times1800\times1800$, to obtain the critical exponents for a universality class with no previously reported estimates in the literature, namely the replica-like limit $n\rightarrow1$ of $\mathrm{RP}^{n-1}$ sigma field theory. Estimates of critical exponents include $ν=0.918(5)$ and $η=-0.091(9)$. We also study the scaling dimension of the 4-leg watermelon correlators, particularly for $n=1$ we obtain the value $x_4=1.292(8)$.

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One-dimensional Ising spin-glass model with long-range interactions

M A Moore

An Ising chain with Hamiltonian H=-J Sigma i<j epsilon ij S i S j / mod i-j mod (1+ sigma )/2 is considered where the epsilon ij are independent random variables. According to Kotliar et al. (1983) this model has a phrase transition with non-mean-field exponents when 1 / 3 < sigma <1 and mean-field exponents beyond the upper critical value of sigma = 1 / 3 . By means of an epsilon expansion about the lower critical value of sigma =1, the model is shown to be replica symmetric as epsilon to 0 and it is speculated that this result holds for 1 / 3 < sigma <1.

Journal of Physics A Mathematical and General•Mar 11, 1986•11 citations
3D Unoriented loop models and the RPn−1\mathrm{RP}^{n-1}RPn−1 sigma model
We study a completely-packed loop model with crossings in a three-dimensional lattice and confirm it is described by RPn−1\mathrm{RP}^{n-1}RPn−1 sigma field theories. We use Monte Carlo simulations, with systems sizes up to 1800×1800×18001800\times1800\times18001800×1800×1800, to obtain the critical exponents for a universality class with no previously reported estimates in the literature, namely the replica-like limit n→1n\rightarrow1n→1 of RPn−1\mathrm{RP}^{n-1}RPn−1 sigma field theory. Estimates of critical exponents include ν=0.918(5)ν=0.918(5)ν=0.918(5) and η=−0.091(9)η=-0.091(9)η=−0.091(9). We also study the scaling dimension of the 4-leg watermelon correlators, particularly for n=1n=1n=1 we obtain the value x4=1.292(8)x_4=1.292(8)x4​=1.292(8).
An Ising chain with Hamiltonian H=-J Sigma i<j epsilon ij S i S j / mod i-j mod (1+ sigma )/2 is considered where the epsilon ij are independent random variables. According to Kotliar et al. (1983) this model has a phrase transition with non-mean-field exponents when 1 / 3 < sigma <1 and mean-field exponents beyond the upper critical value of sigma = 1 / 3 . By means of an epsilon expansion about the lower critical value of sigma =1, the model is shown to be replica symmetric as epsilon to 0 and it is speculated that this result holds for 1 / 3 < sigma <1.